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How to use the rule of three — step by step (with a worked example)

Q: What is the difference between direct and inverse?

Direct means more leads to more — both quantities grow in the same ratio, and you work through the value-for-one. Inverse means more leads to less — the product of the two quantities stays constant instead.

Q: How do I know it is inversely proportional?

Look at how the quantities move: if one rises while the other falls, it is inversely proportional. For example, more workers means less time. If both rise together, it is direct.

Q: How do you work the inverse rule of three?

Multiply the two known values into a constant product, then divide by the new amount. If 4 workers take 6 hours, that is 4 · 6 = 24 worker-hours, so 3 workers finish in 24 ÷ 3 = 8 hours.

Q: What is the rule of three used for?

For percentages, scales, mixtures, speeds and price comparisons — anywhere two quantities are proportional to each other and a fourth value is wanted.

Q: Can I use the rule of three with decimals?

Yes, the method is exactly the same. It produces values-for-one such as 0.02 L per km or 4.5 € per metre. The calculator accepts a dot or a comma as the decimal separator.

The rule of three finds the fourth value of a proportion when you already know three. In a direct proportion you work through the value for one unit; in an inverse proportion you work through a constant product. Worked example: 3 kg cost 12 €, what do 5 kg cost? → 20 €. Suitable for Grade 6–7 / Year 7–8.

By Valerij Hofmann · Fullstack Developer

Updated May 29, 2026Beginner

Quick answer

For a direct proportion, scale down to one unit, then up to the amount you want: if 3 kg cost 12 €, then 1 kg costs 12 ÷ 3 = 4 €, so 5 kg cost 5 · 4 = 20 €. For an inverse proportion (more means less), the product of the two quantities stays constant.

At a glance

Summary of this tutorial
Example	3 kg → 12 €, 5 kg → ?
Method	Direct rule of three (value for 1)
Steps	4
Answer	20 €
Inverse	Product constant, then divide
Grade level	Grade 6–7 (ages 11–13)

Worked example: 3 kg → 12 €, 5 kg → ?

EXAMPLE

3 kg → 12 €, 5 kg → ?

Directly proportional: more kilograms means more euros. We work through the price for 1 kg.

How to do the rule of three step by step

These steps work for any directly proportional pairing of the form a → b, c → ?.

Step 1 · Start
3 kg → 12 €
The known pairing: 3 kg cost 12 €.
Step 2 · ÷3
1 kg → 12 ÷ 3 = 4 €
Scaling down to one unit gives the price for 1 kg.
Step 3 · ×5
5 kg → 5 · 4
Multiply by the amount you want.
Step 4 · Result
= 20 €
5 kg cost 20 €.

Why the rule of three works

In a direct proportion the ratio of value to amount is always the same: 12 € over 3 kg is 4 € per kg, just like 20 € over 5 kg. Going through the value for one unit makes this fixed ratio usable. In an inverse proportion it is the product that stays constant instead — 4 workers · 6 hours = 24 worker-hours stays 24 no matter how many workers join in.

Practice it yourself

Open the rule of three calculator

Enter your own values. Direct or inverse. See every step.

Start practice set

Rule of three questions

Frequently asked questions

How do you solve “3 kg cost 12 €, what do 5 kg cost?”

First scale down to one unit: 1 kg costs 12 ÷ 3 = 4 €. Then scale up to the amount you want: 5 kg cost 5 · 4 = 20 €. That is the direct rule of three through the value for one unit.

What is the difference between direct and inverse?

How do I know it is inversely proportional?

How do you work the inverse rule of three?

What is the rule of three used for?

Can I use the rule of three with decimals?

Quick answer

At a glance

Worked example: 3 kg → 12 €, 5 kg → ?

How to do the rule of three step by step

Step 1 · Start

Step 2 · ÷3

Step 3 · ×5

Step 4 · Result

Why the rule of three works

Practice it yourself

Frequently asked questions